Mathematical measures of societal polarisation

In opinion dynamics, as in general usage, polarisation is subjective. To understand polarisation, we need to develop more precise methods to measure the agreement in society. This paper presents four mathematical measures of polarisation derived from graph and network representations of societies and information-theoretic divergences or distance metrics. Two of the methods, min-max flow and spectral radius, rely on graph theory and define polarisation in terms of the structural characteristics of networks. The other two methods represent opinions as probability density functions and use the Kullback–Leibler divergence and the Hellinger distance as polarisation measures. We present a series of opinion dynamics simulations from two common models to test the effectiveness of the methods. Results show that the four measures provide insight into the different aspects of polarisation and allow real-time monitoring of social networks for indicators of polarisation. The three measures, the spectral radius, Kullback–Leibler divergence and Hellinger distance, smoothly delineated between different amounts of polarisation, i.e. how many cluster there were in the simulation, while also measuring with more granularity how close simulations were to consensus. Min-max flow failed to accomplish such nuance.

Theorem 1. Let A be a square block diagonal matrix consisting of m square matrices of ones J ni where n i is the dimension of the matrix of ones i, and i = 1, . . . , m. Then the spectral radius ρ(A) is equal to the dimension of the largest unit matrix in A.
Proof. Since A is block diagonal, the eigenvalues of A are the eigenvalues of J n1 , . . . , J nm . We know that for a general square matrix of ones J k [1] its characteristic equation is 0 = (k − λ) λ k−1 .
The dimensional values n 1 , . . . , n m are all eigenvalues of their respective matrix including multiple eigenvalues equal to 0. Then n 1 , . . . , n m must be eigenvalues of A, the largest of which is ρ(A), which is also the dimension of the largest matrix of ones in A.
S3 Appendix. Alternate min-max flow for HK bounded confidence. To simplify the calculation of the min-max flow rate for the HK bounded confidence model, one might consider the following method: 1. For every agent in a simulation, count the number of agents within of the agent's opinion.
2. Minimise over those agent counts.
Such a method counts the degree of each agent if you were to develop an adjacency matrix at that specific time in the simulation. Intuitively the method should be identical to min-max flow, but it produces different results when a simulation enters polarisation (see S1 Fig for an example). This method converges to the smallest cluster size instead of reaching zero. We thus consider this method distinct from the min-max flow method.
S4 Appendix. Derivation of the KLD of two normal distributions. Let X ∼ N (µ i , σ i 2 ) and Y ∼ N (µ j , σ j 2 ), and f (x) and g(y) describe the probability density function for X and Y then S5 Appendix. Derivation of the H-distance of two uniform distributions. Let x 1 and x 2 be the centers of two uniform distributions f (x) and g(x) both with width 2 and, without loss of generality, let x 1 > x 2 . The Hellinger distance is There is two distinct cases for the Hellinger affinity. First is when there is no overlap, i.e. x 1 − x 2 > 2 , between f and g which means that the Hellinger affinity is zero and hence Second is when there is overlap, i.e. x 1 − x 2 ≤ 2 , and the Hellinger affinity is non-zero. Specifically the Hellinger affinity will be the area of the overlap which is We can conclude that S6 Appendix. Estimating cluster count from exponential, mean KLD growth. Consider a Martins simulation that has reached steady-state, let Ω be the set of all agents in the simulation, and the simulation has divided into ψ separate opinion clusters such that where A k is a set of agents in the kth opinion cluster such that In steady-state, all agents in a Martins simulations have σ → 0 . Then, according to Eq 5, agents will only achieve a p * = 1 when x i = x j i.e. when two agent are in the same cluster. If x i = x j i.e. when two agent are in the different clusters, then, with σ → 0, p * = 0. Also, from Eq 5, when p * = 1, agents will halve their σ 2 , which means that Eq 7 will double for a select proportion of agent pairs in the simulation, but not for all agent pairs. Meaning that KLD will proportionally grow by a fixed amount a after a single p * = 1 interaction where t is an arbitrary number of interactions after the simulation has reached 591 steady-state and s is the number of interactions until a p * = 1 interaction occurs.

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In a single p * = 1 interaction, two agents will halve their 'variance' (uncertainty squared), doubling the KLD between those agents and every other agent in the simulation. All other pairings will maintain the same KLD. The proportional growth of KLD, a, is dependent only on the pairwise agent's KLD where KLD = 0. We refer to agent pairings that have a KLD = 0 as non-contributing and those with KLD > 0 as contributing. Therefore, a = (1 − q) + 2q = 1 + q where q is the proportion of contributing pairings that double their KLD. Let n be the number of agents in the simulation and consider an agent inside a cluster, only n(1 − 1/ψ) agents would generate contributing KLDs since agents inside the hypothetical agent's cluster would generate a KLD = 0. Since two agents will be interacting, we can double this agent count to get the total number KLDs that double from two opinions updating, resulting in 2n(1 − 1/ψ). The total number of contributing pairings will be the total number of possible pairings, n 2 , minus the non-contributing pairings, i.e. pairings which pair agents from the same clusters, ψ(n/ψ) 2 . Therefore the total number of contributing pairings is n 2 (1 − 1/ψ). It follow then that q = 2/n. Thus a = n + 2 n ,